The above SOCP problem can be solved by the interior point method [1]}.
Now, write \(w=w_0+Fu\)  with \(w_0=({\mu _P-\mu _N}){\Vert \mu _P-\mu _N\Vert _2^{-2}}\)  and \(F\in {\mathbb {R}}^{p\times (p-1)}\) 
the orthogonal matrix whose columns span the subspace of the vectors orthogonal to \((\mu _P-\mu _N)\) .
Then the constraint can be eliminated to get the unconstrained SOCP [2]}:
\(\min _{u\ne 0}\Vert \Sigma _P^{1/2}(w_0+F u)\Vert _2+\Vert \Sigma _N^{1/2}(w_0+F u)\Vert _2,\) 
